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Remarks on the symbolic calculus in vector valued Besov spaces. (Remarques sur le calcul symbolique dans certains espaces de Besov à valeurs vectorielles.) (French) Zbl 1182.46019

Summary: We are interested in the superposition operators \(T_{ f }(g):=f\circ g\) on vector valued Besov and Lizorkin-Triebel spaces of positive smoothness exponent \(s\). As a first step towards the characterization of functions which operate, we establish that the local Lipschitz continuity of \(f\) is necessary if the space \(B_{ p,q}^s(\mathbb R^{{ n }},\mathbb R{^{ m }})\) or \(F_{ p,q}^s(\mathbb R{^{ n }},\mathbb R{^{ m }})\) is imbedded into \(L_{ \infty }(\mathbb R^{{ n }},\mathbb R^{ m })\), and that the uniform Lipschitz continuity of \(f\) is necessary if the space is not imbedded into \(L_{ \infty }(\mathbb R{^{ n }},\mathbb R^{ m })\). We also prove that the local membership to the same space is necessary for \(m\leq n\). We finally study the regularity of the superposition operator \(T_{ f }\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
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References:

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